Problem: Simplify the following expression: $z = \dfrac{-30n^3 - 21n^2}{-30n^3 + 18n^2}$ You can assume $n \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-30n^3 - 21n^2 = - (2\cdot3\cdot5 \cdot n \cdot n \cdot n) - (3\cdot7 \cdot n \cdot n)$ The denominator can be factored: $-30n^3 + 18n^2 = - (2\cdot3\cdot5 \cdot n \cdot n \cdot n) + (2\cdot3\cdot3 \cdot n \cdot n)$ The greatest common factor of all the terms is $3n^2$ Factoring out $3n^2$ gives us: $z = \dfrac{(3n^2)(-10n - 7)}{(3n^2)(-10n + 6)}$ Dividing both the numerator and denominator by $3n^2$ gives: $z = \dfrac{-10n - 7}{-10n + 6}$